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Extra resources for 101 Investment Tools for Buying Low and Selling High

Example text

We implicitly assume that averages under the over-line are conditional on µ. 2 2 P F ( Q|µ ). µ=1 = Q 2. Generic behaviour of the Minority Game Let us rephrase in mathematical terms the generic behaviour of the Minority Game sketched in the previous chapter. Early papers focused on the cooperative properties of the system in the stationary state. Symmetry arguments suggest that none of the two groups 0 or 1 will be systematically the minority one. This means that A(t) will 9 That a stationary state exists for the Minority Game exists can be shown observing that (i) it is a Markov process (ii) it can be approximated with a ﬁnite space Markov process (discretizing Yi (t) on a grid of points) to an arbitrary precision (letting → ∞).

What opened up before our eyes was a dazzling parade of new phenomena, unheard of in our previous complexity modelling. Our ﬁrst concern was to understand how players’ information-processing capacity, here conveniently represented by the memory length M, impacted on their performance. It was also feasible to check how the size of the ‘idea-bag’—that is, the number of alternative strategies each player possesses—affects performance. What we were most curious about, motivated by real markets, was to see how players of different information capacities fare when forced to cohabit.

The extraction of quantitative results from this approach is made difﬁcult by the mathematical complexity of the resulting equations. On the other hand, this method is rigorous and exact. Even if it does not yet provide a complete picture of how Minority Game works, the generating function is a very promising approach. We focus on the simplest non-trivial version of the model to make the derivation as straight and simple as possible. The derivation can be extended in a straightforward manner to a number of variations on the theme, adding frills here and there on the skeleton Minority Game which we discuss.